Dissipative Preparation of Spin Squeezed Atomic Ensembles in a Steady State

We present and analyze a new approach for the generation of atomic spin-squeezed states. Our method involves the collective coupling of an atomic ensemble to a decaying mode of an open optical cavity. We demonstrate the existence of a collective atomic dark state, decoupled from the radiation field. By explicitly constructing this state we find that it can feature spin squeezing bounded only by the Heisenberg limit. We show that such dark states can be deterministically prepared via dissipative means, thus turning dissipation into a resource for entanglement. The scaling of the phase sensitivity taking realistic imperfections into account is discussed.

Figure 1

(a) Schematic experimental setup of atoms with a spontaneous emission rate $\gamma$ in a cavity with linewidth $\kappa$ supporting a single cavity mode $a$. (b) Effective linkage pattern consisting of two degenerate ground states ($|\pm ⟩$) encoding the effective atomic spin and two excited states $|e_{\pm }⟩$. The system is driven via classical control fields ${\Omega }_{\pm }$ with large detuning $\Delta$ and a single global cavity mode $a$ coupling homogeneously to all atoms. The orientation of the arrows does not necessarily correspond to the actual polarization of the fields in a specific physical realization. We give an example of such an implementation in the Supplemental Material [29].

Figure 2

Phase sensitivity of the dark state $|{\psi }_{\mathrm{spin}}⟩$. Exact numerical solution (solid blue curve) and mean-field approximation (dashed red curve) for $N=10$ (left) and $N=100$ (right) atoms. The optimal squeezing is obtained for ${\Omega }_{+}\to {\Omega }_{-}$ and approaches the Heisenberg limit $\delta \varphi =1/\sqrt{N(N/2+1)}\sim 1/N$.

Figure 3

Time evolution of the phase sensitivity for $N=10$ spins and ${\Omega }_{-}/{\Omega }_{+}=0.2$. Numerical solution of the master equation, with fully polarized (solid curves) or randomly generated (dash-dotted curves) initial state, and mean-field approximation (dashed curves). (a) The atoms are coupled only to the cavity, i.e., $\gamma =0$. (b) Finite scattering rate $\gamma \ne 0$ with single-atom cooperativity $\chi =1$. Observe that the steady state does not depend on the initial state.

Figure 4

Optimal phase sensitivity in the steady state, in the presence of single spin decay for $N={10}^6$. The solid curve is given by the numerical minimization of Eqs. (14, 15). The dash-dotted and dashed curves are Eqs. (16, 17), respectively.